Zhibo Chen

On polynomial functions from Z n to Z m

Abstract We define the concept of a polynomial function from Z n to Z m , which is a generalization of the well-known polynomial function from Z n to Z m . We obtain a necessary and sufficient condition on n and m for all functions from Z n to Z m to be polynomial functions. Then we present canonical representations and the counting formula for the polynomial functions from Z n to Z m . Further, we give an answer to the following problem: How to determine whether a given function from Z n to Z m is a polynomial function, and how to obtain a polynomial to represent a polynomial function?

On polynomial functions from Z n 1 ×Z n 2 × l×Z n r to Z m

Abstract The well-known concept of a polynomial function (mod m ) has been generalized to polynomial functions from Z n to Z m , and a number of results have been obtained in (Chen, 1995). In the present paper, we further define the concept of polynomial functions from Z n 1 × Z n 2 × … × Z n r to Z m , and generalize the results of (Chen, 1995). We give a canonical representation and the counting formula for such polynomial functions. Then we obtain a necessary and sufficient condition on n 1 , n 2 , …, n r and m for all functions from Z n 1 × Z n 2 × … × Z n r to Z m to be polynomial functions. Further, we give an answer to the following problem: How to determine whether a given function from Z n 1 × Z n 2 × … × Z n r to Z m is a polynomial function, and how to obtain a polynomial to represent a polynomial function from Z n 1 × Z n 2 × … × Z n r to Z m ?

Integral sum graphs from identification

The idea of integral sum graphs was introduced by Harary (1994). A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u) + f(v) = f(w) for some node w in G. A tree is said to be a generalized star if it can be obtained from a star by extending each edge to a path. A node of a tree T is said to be a fork of T if its degree is not equal to two. In this paper, we first introduce some methods of identification on constructing new connected integral sum graphs from given integral sum graphs. Applying the methods of identification, we then prove that the generalized stars and the trees with all forks at least distance 4 apart are integral sum graphs.

Harary's conjectures on integral sum graphs

Abstract Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N(Z) is the graph (S, E) with uv ϵ E if and only if u + v ϵ S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂ N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ⩾ 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.

Ordering graphs with small index and its application

We consider the problem of ordering connected graphs by index (the largest eigenvalue). The asymptotic ordering for the connected graphs with index less than √2+√5 is determined. Its application to the study of acyclic Kekulean molecules with big HOMO-LUMO separation is also given.

On k-pairable graphs

Motivated by a 1994 result of Graham et al. (Amer. Math. Monthly 101(7) (1994) 664) about spanning trees of the graphs with an antipodal isomorphism, we introduce the concept of k-pairable graphs and extend the result in Graham et al. (Amer. Math. Monthly 101(7) (1994) 664) to this larger class of graphs. We then define a new graph parameter p(G), called the pair length of graph G. This parameter measures the maximum distance, in some sense, between a subgraph induced by half the vertices of G with the isomorphic subgraph induced by the other half of V(G). An upper bound for the parameter p(G) is given. Some properties of the k-pairable graphs and their product graphs are studied. We also post some problems for further research.

Forcing faces in plane bipartite graphs

Let @W denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph [emailxa0protected][emailxa0protected] is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n>=4 and n>=k>=0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.

On integral sum graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v)=f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K3 has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous related results are also presented.

On k-pairable regular graphs

Let k be a positive integer. A graph G is said to be k-pairable if its automorphism group contains an involution @f such that d(x,@f(x))>=k for any vertex x of G. The pair length of a graph G, denoted as p(G), is the maximum k such that G is k-pairable; p(G)=0 if G is not k-pairable for any positive integer k. Some new results have been obtained since these concepts were introduced by Chen [Z. Chen, On k-pairable graphs, Discrete Mathematics 287 (2004) 11-15]. In the present paper, we first introduce a new concept called strongly induced cycle and use it to give a condition for a graph G to have p(G)=k. Then we consider the class G(r,k) of prime graphs which are r-regular and have pair length k. For any integers r,k>=2, except r=k=2, we show that the set G(r,k) is not empty, determine the minimum order of a graph in G(r,k), and give a construction for such a graph with the minimum order. With this approach, we also obtain the minimum order of an r-regular graph with pair length k for any integers r,k>=2. Finally, we post an open question for further research.

Forcing faces in plane bipartite graphs (II)

The concept of forcing faces of a plane bipartite graph was first introduced in Che and Chen (2008) [3] [Z. Che, Z. Chen, Forcing faces in plane bipartite graphs, Discrete Mathematics 308 (2008) 2427-2439], which is a natural generalization of the concept of forcing hexagons of a hexagonal system introduced in Che and Chen (2006) [2] [Z. Che and Z. Chen, Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (2006) 649-668]. In this paper, we further extend this concept from finite faces to all faces (including the infinite face) as follows: A face s (finite or infinite) of a 2-connected plane bipartite graph G is called a forcing face if the subgraph G-V(s) obtained by removing all vertices of s together with their incident edges has exactly one perfect matching. For a plane elementary bipartite graph G with more than two vertices, we give three necessary and sufficient conditions for G to have all faces forcing. We also give a new necessary and sufficient condition for a finite face of G to be forcing in terms of bridges in the Z-transformation graph Z(G) of G. Moreover, for the graphs G whose faces are all forcing, we obtain a characterization of forcing edges in G by using the notion of handle, from which a simple counting formula for the number of forcing edges follows.